Max-linear systems: theory and algorithms
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London [u.a.]
Springer
2010
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| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 261 - 267 |
| Physical Description: | XVII, 272 S. graph. Darst. |
| ISBN: | 9781849962988 |
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Record in the Search Index
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| adam_text | Titel: Max-linear Systems
Autor: Butkovič, Peter
Jahr: 2010
Contents
Introduction................................ 1
1.1 Notation. Definitions and Basic Properties............. 1
1.2 Examples............................... 7
1.3 Feasibility and Reachability..................... 9
1.3.1 Multi-machine Interactive Production Process:
A Managerial Application.................. 9
1.3.2 MMIPP: Synchronization and Optimization ........ 10
1.3.3 Steady Regime and Its Reachability............. 11
1.4 About the Ground Set........................ 12
1.5 Digraphs and Matrices........................ 13
1.6 The Key Players........................... 16
1.6.1 Maximum Cycle Mean ................... 17
1.6.2 Transitive Closures ..................... 21
1.6.3 Dual Operators and Conjugation .............. 29
1.6.4 The Assignment Problem and Its Variants.......... 30
1.7 Exercises............................... 36
Max-algebra: Two Special Features................... 41
2.1 Bounded Mixed-integer Solution to Dual Inequalities:
A Mathematical Application..................... 41
2.1.1 Problem Formulation .................... 41
2.1.2 All Solutions to SDI and All Bounded Solutions...... 42
2.1.3 Solving BMISDI....................... 43
2.1.4 Solving BMISDI for Integer Matrices............ 45
2.2 Max-algebra and Combinatorial Optimization ........... 48
2.2.1 Shortest/Longest Distances: Two Connections....... 48
2.2.2 Maximum Cycle Mean ................... 49
2.2.3 The Job Rotation Problem.................. 49
2.2.4 Other Problems ....................... 51
2.3 Exercises............................... 52
One-sided Max-linear Systems and Max-algebraic Subspaces .... 53
3.1 The Combinatorial Method..................... 53
3.2 The Algebraic Method........................ 57
3.3 Subspaces, Generators, Extremals and Bases............ 59
3.4 Column Spaces ........................... 64
3.5 Unsolvable Systems......................... 67
3.6 Exercises............................... 69
Eigenvalues and Eigenvectors...................... 71
4.1 The Eigenproblem: Basic Properties ................ 71
4.2 Maximum Cycle Mean is the Principal Eigenvalue......... 74
4.3 Principal Eigenspace......................... 76
4.4 Finite Eigenvectors ......................... 82
4.5 Finding All Eigenvalues....................... 86
4.6 Finding All Eigenvectors ...................... 95
4.7 Commuting Matrices Have a Common Eigenvector ........ 97
4.8 Exercises............................... 98
Maxpolynomials. The Characteristic Maxpolynomial......... 103
5.1 Maxpolynomials and Their Factorization.............. 105
5.2 Maxpolynomial Equations...................... Ill
5.3 Characteristic Maxpolynomial.................... 112
5.3.1 Definition and Basic Properties............... 112
5.3.2 The Greatest Corner Is the Principal Eigenvalue...... 114
5.3.3 Finding All Essential Terms of a Characteristic
Maxpolynomial....................... 116
5.3.4 Special Matrices....................... 123
5.3.5 Cayley-Hamilton in Max-algebra ............. 124
5.4 Exercises............................... 126
Linear Independence and Rank. The Simple Image Set........ 127
6.1 Strong Linear Independence..................... 127
6.2 Strong Regularity of Matrices.................... 130
6.2.1 A Criterion of Strong Regularity .............. 130
6.2.2 The Simple Image Set.................... 135
6.2.3 Strong Regularity in Linearly Ordered Groups....... 137
6.2.4 Matrices Similar to Strictly Normal Matrices........ 138
6.3 Gondran-Minoux Independence and Regularity .......... 138
6.4 An Application to Discrete-event Dynamic Systems........ 144
6.5 Conclusions............................. 146
6.6 Exercises............................... 146
Two-sided Max-linear Systems ..................... 149
7.1 Basic Properties........................... 150
7.2 Easily Solvable Special Cases.................... 151
7.2.1 A Classical One....................... 151
7.2.2 Idempotent Matrices..................... 152
7.2.3 Commuting Matrices .................... 153
7.2.4 Essentially One-sided Systems ............... 153
7.3 Systems with Separated Variables-The Alternating Method . ... 156
7.4 General Two-sided Systems..................... 162
7.5 The Square Case: An Application of Symmetrized Semirings ... 164
7.6 Solution Set is Finitely Generated.................. 169
7.7 Exercises............................... 176
8 Reachability of Eigenspaces.......................179
8.1 Visualization of Spectral Properties by Matrix Scaling.......181
8.2 Principal Eigenspaces of Matrix Powers ..............186
8.3 Periodic Behavior of Matrices....................188
8.3.1 Spectral Projector and the Cyclicity Theorem........ 188
8.3.2 Cyclic Classes and Ultimate Behavior of Matrix Powers . . 193
8.4 Solving Reachability......................... 196
8.5 Describing Attraction Spaces.................... 202
8.5.1 The Core Matrix....................... 203
8.5.2 Circulant Properties..................... 204
8.5.3 Max-linear Systems Describing Attraction Spaces..... 206
8.6 Robustness of Matrices ....................... 212
8.6.1 Introduction......................... 212
8.6.2 Robust Irreducible Matrices................. 213
8.6.3 Robust Reducible Matrices................. 215
8.6.4 M-robustness ........................ 220
8.7 Exercises............................... 223
9 Generalized Eigenproblem........................227
9.1 Basic Properties of the Generalized Eigenproblem.........228
9.2 Easily Solvable Special Cases....................230
9.2.1 Essentially the Eigenproblem................230
9.2.2 When A and B Have a Common Eigenvector........230
9.2.3 When One of A, B Is a Right-multiple of the Other .... 231
9.3 Narrowing the Search for Generalized Eigenvalues.........233
9.3.1 Regularization........................ 233
9.3.2 A Necessary Condition for Generalized Eigenvalues .... 234
9.3.3 Finding maper|C(A.)| .................... 235
9.3.4 Narrowing the Search.................... 236
9.3.5 Examples........................... 238
9.4 Exercises............................... 241
10 Max-linear Programs...........................243
10.1 Programs with One-sided Constraints................243
10.2 Programs with Two-sided Constraints................245
10.2.1 Problem Formulation and Basic Properties.........245
10.2.2 Bounds and Attainment of Optimal Values.........247
10.2.3 The Algorithms....................... 251
10.2.4 The Integer Case....................... 253
10.2.5 An Example......................... 255
10.3 Exercises............................... 257
11 Conclusions and Open Problems.................... 259
References................................... 261
Index...................................... 269
|
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| discipline | Mathematik |
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| id | DE-604.BV040261743 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:20:15Z |
| institution | BVB |
| isbn | 9781849962988 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025117506 |
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| physical | XVII, 272 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
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| publisher | Springer |
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| series2 | Springer monographs in mathematics |
| spelling | Butkovič, Peter Verfasser aut Max-linear systems theory and algorithms Peter Butkovič London [u.a.] Springer 2010 XVII, 272 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Literaturverz. S. 261 - 267 Mathematik Matrix theory Mathematics Matrix Mathematik (DE-588)4037968-1 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 s DE-604 Erscheint auch als Online-Ausgabe 978-1-8499-6299-5 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025117506&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Butkovič, Peter Max-linear systems theory and algorithms Mathematik Matrix theory Mathematics Matrix Mathematik (DE-588)4037968-1 gnd |
| subject_GND | (DE-588)4037968-1 |
| title | Max-linear systems theory and algorithms |
| title_auth | Max-linear systems theory and algorithms |
| title_exact_search | Max-linear systems theory and algorithms |
| title_full | Max-linear systems theory and algorithms Peter Butkovič |
| title_fullStr | Max-linear systems theory and algorithms Peter Butkovič |
| title_full_unstemmed | Max-linear systems theory and algorithms Peter Butkovič |
| title_short | Max-linear systems |
| title_sort | max linear systems theory and algorithms |
| title_sub | theory and algorithms |
| topic | Mathematik Matrix theory Mathematics Matrix Mathematik (DE-588)4037968-1 gnd |
| topic_facet | Mathematik Matrix theory Mathematics Matrix Mathematik |
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